From Elroch's good postings - caused me to look up Fermat.
"Fermat's principle states that “light travels between two points along the path that requires the least time, as compared to other nearby paths.” From Fermat's principle, one can derive (a) the law of reflection [the angle of incidence is equal to the angle of reflection] and (b) the law of refraction [Snell's law]."
Pierre de Fermat was a genius.
A peer of Descartes and Pascal. In communication with Pascal.
Apparently in France in those days - first half of the 1600s - a lot of mathematics was 'secret'.
After Fermat - Newton and Leibnitz and Euler - a Swiss.
For several hundreds of years everyone great in math and science was born and lived and operated with a short radius from Paris - whether in the south of france and Switzerland - Austria and Germany - Belgium and Holland - England and Scotland.
Later expanded to include the US. Franklin.
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Fermat's last thereom ...
"In 1637 the French mathematician Pierre de Fermat wrote in his copy of the Arithmetica by Diophantus of Alexandria (c. 250 CE), “It is impossible for a cube to be a sum of two cubes, a fourth power to be a sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.” For centuries mathematicians were baffled by this statement, for no one could prove or disprove Fermat’s last theorem."
Apparently Fermat's last thereom was recently proven by Andrew Wiles.
A professor at Oxford University. Alive and kicking in his early seventies.
That figures. Pun intended.
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Could there be confusion between Fermat's Last Theorem and his 'Little Theorem' and Euler's Little Theorem?
Try yes.
But - a way to study math and science ...
Study the greats and their timelines.
What understanding led to the next understanding and so on.
Find out how formulas were derived instead of trying to just memorize them.
Just memorize could lead to just mesmerize. Mesmer. He messed up. Was refuted.
Chess will never be solved, here's why
btw fermat's last theorem should (IMO)be called wiles's theorem, the story that wiles went down to prove it is absolutely insane, and his name absolutely deserves to be recognized.
@12617
"it is inappropriate/unwise to be certain without being able to justify the certainty"
++ Are you certain the Sun will rise tomorrow and do you have justification?
Are you certain if you drop an apple it will fall towards the Earth and do you have justification?
I am certain chess is a draw with optimal play, and I have justification.
I am certain 1 e4 e5 2 Ba6? loses for white with optimal play and I have justification.
I am certain 1 e4 Nf6 2 Qh5? loses for white with optimal play and I have justification.
I am not certain all 26 ongoing ICCF WC Finals games will end as draws like the 110 that finished.
Probability is linked to an experiment, either real or in thought.
""it is inappropriate/unwise to be certain without being able to justify the certainty"
++ Are you certain the Sun will rise tomorrow and do you have justification?
Are you certain if you drop an apple it will fall towards the Earth and do you have justification?"
does bro not understand what a mathematical proof is? oh wait, we've already long established that you have no idea what it is.
"I am certain chess is a draw with optimal play, and I have justification."
you have corroboration, not justification.
it's either been rigorously mathematically proven from axioms, or it hasnt been.
it's pretty simple.
What tygxc is doing is like an opposite of solipsism.
A solopsist will not let anything be proven to him or her.
But its usually used tactically. In context. As opposed to being a mental disorder.
tygxc doing an opposite -
something mathematical is 'proven' to him without thorough proof and he even extends this to future events.
Slipshod pseudo-analysis 'satisfies' tygxc and then he pushes it too. On the internet.
A kind of disinformation.
But such operations by him are getting interfered with. Rightly.
He seems to like it. Plus the process is an engine of discussion as it were.
btw fermat's last theorem should (IMO)be called wiles's theorem, the story that wiles went down to prove it is absolutely insane, and his name absolutely deserves to be recognized.
Wiles proving Fermat's last thereom is a gigantic thing.
And so recent! And so Long After Fermat!
Over 300 years Later!
They both qualify as great.
@12637
"proof that perfect games exist" ++ Of course perfect games exist, and of course that is proven within Combinatorial game theory.
"wait 1 or a few years, when the next chess playing apparatus beats up on the current best"
++ That is the point: each year there were fewer and fewer decisive games in the ICCF World Championship Finals and now there are none.
@12635
"e.g. Shirov's game I posted where the bishop to rook 6 sacrifice wins"
++ I make no general statements about all sacrifices, I only make specific statements:
1 e4 e5 2 Ba6?, 1 e4 e5 2 Nf3 Nc6 3 Nd4?, 1 e4 e5 2 Nf3 Nc6 3 Nxe5?, 1 e4 e5 2 Nf3 Nc6 3 Ng5?, 1 e4 e5 2 Nf3 Nc6 3 Nh4? all lose for white.
I make no general statements about all knight moves, but 1 Nh3 cannot be better than 1 Nf3 and 1 e4 e5 2 Nf3 Nc6 3 Ng1 cannot be better than 3 Bb5 or 3 Bc4.
'If nobody makes a mistake then a chess game is likely to end in a draw'
if - then.
But there's a Kicker.
A Zinger.
A flip side.
There are mistakes. Constantly.
Including mistakes that today's engines can't even pick up - some of which engines ten years from now Will. Then engines ten years after that - similiar thing.
'best play by both sides'
has such a thing ever happened?
Where? When?
Rephrase:
"If both sides do not make a big enough mistake or mistakes for the other side to catch the mistakes or exploit them sufficiently or the other side simply doesn't do so then the game is likely to end in a draw but that's only provided somebody's clock doesn't go down first which happens much of the time - and then No Draw most of 'the time' then."
is a much better way of putting it.
Because the first way isn't in tune with reality.
The second one is.
@12645
"it not a useful way to bring 10^43 down to 10^17"
++ Bringing down 10^44 to 10^38 is by eliminating vast numbers of positions like the 3 random samples shown in https://github.com/tromp/ChessPositionRanking. None of these can result from optimal play by both sides, as there are underpromotions to rooks and or bishops from both sides and underpromoting to a rook or bishop only makes sense to avoid a draw by stalemate, and it cannot be optimal play for both sides to avoid the draw.
Bringing down 10^38 to 10^34 is by eliminating positions like
10^34 is conservative: Tromp conjectured a reduction of 10^6 i.e. to 10^32.
The reduction from 10^34 to 10^17 is the square root assuming perfect alpha-beta pruning. Schaeffer arrived only at exponent 0.67 for Checkers, but chess programs have evolved more than Chinook, and Chess is easier to prune than Checkers.
The real exponent lies somewhere between 0.5 and 0.67.
"You need to be able to discard entire classes of positions"
++ Yes: after 1 e4 discard all classes of positions with a white pawn on e2.
After 1 e4 c5 discard all classes of positions with a black pawn on c7.
After 1 e4 c5 2 Nf3 d6 discard all classes of positions with a black pawn on d7.
After 1 e4 c5 2 Nf3 d6 3 d4 discard all classes of positions with a white pawn on d2.
After 1 e4 c5 2 Nf3 d6 3 d4 cxd4 discard all classes of positions with 32 men, or 8 white pawns.
After 1 e4 c5 2 Nf3 d6 3 d4 cxd4 4 Nxd4 discard all classes of positions with 31 men, or 8 black pawns.
After 1 e4 c5 2 Nf3 d6 3 d4 cxd4 4 Nxd4 Nf6 5 Nc3 a6 discard all classes of positions with a black pawn on a7.
"any position with 3 bishops for one player can be discarded"
++ The only way to reach 3 bishops is by underpromoting a pawn. The only reason to underpromote to a bishop is to avoid a draw by stalemate. It happens, but it is very rare.
By the time a pawn reaches the back rank normally one bishop has been traded.
It happens that a pawn promotes with 2 bishops still on the board, but it is very rare.
The combination of both unlikely events: underpromotion to a bishop to avoid stalemate,
and a pawn reaching the back rank with both bishops still on the board never occurs.
"any position with 3 bishops for one player can be discarded"
++ The only way to reach 3 bishops is by underpromoting a pawn. The only reason to underpromote to a bishop is to avoid a draw by stalemate. It happens, but it is very rare.
By the time a pawn reaches the back rank normally one bishop has been traded.
It happens that a pawn promotes with 2 bishops still on the board, but it is very rare.
The combination of both unlikely events: underpromotion to a bishop to avoid stalemate,
and a pawn reaching the back rank with both bishops still on the board never occurs.
Arithmetic correction: when you multiply two small numbers, you don't get zero. And it's not even as if they are that small - I am aware of two examples from master praxis where an underpromotion to bishop was the only winning move. This indicates it is more like a one in a million thing than a really unlikely one. 10^44 is more than the 7th power of million. If we had a trillion master games to look it, we wouldn't be surprised to find several multiple underpromotions to bishop and room in different combos. And so on,
We should be in agreement that it is a provable fact that if there is a drawing strategy for a side, there is one without any underpromotion to rook or bishop by that side. And if the opponent underpromotes to bishop or rook, you can use the same moves as you would have if they had promoted to a queen (dealing with moves by the bishop/rook as if the piece was a queen) and the worst that can happen is that you stalemate the opponent (because a move they would have had with a promoted queen is not available when the promoted piece was a bishop or rook). So it suffices to generate drawing strategies for the similar game to chess where underpromotions to bishop and rook are forbidden (but any number of promotions to queen or knight by both sides are permitted).
But here note that you are under a major misapprehension. The number of legal chess positions without ANY promotion is around 4 x 10^37. The appropriate number for drawing strategies is much bigger because both sides can make up to 8 promotions to queen or to knight, or a mixture. You can be sure this adds a large factor to the number.
Note that it is not proven, merely strongly believed that chess is a draw (the certainty of some people who are not good at dealing with uncertainty notwithstanding). If one side has a winning strategy, it is certainly necessary to deal separately with all underpromotions by the opponent to bishop and rook, since these may create stalemate possibilities. In addition any number of each type of underpromotion could be needed in a line of the winning strategy, for the same reason.
It's safe to say chess still can't be rigorously solved. If you want to do it unrigorously, any guy claiming it is a draw will do fine.
[Also, the final version of Chinook may itself have been unbeatable. Today's top chess engines will probably lose matches to those of the future. So no, it is not reasonable to claim they are more advanced relative to the complexity of the game]
@12647
"I am aware of two examples from master praxis where an underpromotion to bishop was the only winning move." ++ So that makes 1 position in 10^6 games.
Now of how many examples from master praxis are you aware where a pawn promoted and both bishops were still on the board? ++ That is maybe also 1 position in 10^6 games.
That gives 1 position in 10^17 positions. To estimate the number of relevant positions to weakly solve Chess (10^17) that is negligible indeed.
"So it suffices to generate drawing strategies for the similar game to chess where underpromotions to bishop and rook are forbidden" ++ No they are allowed, but underpromotions to pieces not previously captured not.
"The number of legal chess positions without ANY promotion is around 4 x 10^37."
++ Correction: without any promotion to a piece not previously captured.
For brevity the article title does not mention that, but it is obvious.
"strongly believed that chess is a draw" ++ To me it is proven.
"It's safe to say chess still can't be rigorously solved."
++ No, on the contrary it is safe to say chess can be rigourously weakly solved.
The 17 ICCF WC Finalist and their servers are doing it now.
110 games out of 110 that redundantly link the initial position in average 39 moves to certain draws.
"Today's top chess engines will probably lose matches to those of the future."
++ At short time control. In TCEC they have to impose 50 unbalanced openings to avoid all draws and they do hit the 7-men endgame table base.
@12647
"I am aware of two examples from master praxis where an underpromotion to bishop was the only winning move." ++ So that makes 1 position in 10^6 games.
Now of how many examples from master praxis are you aware where a pawn promoted and both bishops were still on the board? ++ That is maybe also 1 position in 10^6 games.
Guessing - your modus operandi - is not good enough. This is a very reckless guess, without even shoddy reasoning behind it.That gives 1 position in 10^17 positions.
Based on a made up number, so worthless.
To estimate the number of relevant positions to weakly solve Chess (10^17)
Another made-up number where the reasoning never came close to making sense.
that is negligible indeed.
As well as guessing, you are indeed big on neglecting things.
"So it suffices to generate drawing strategies for the similar game to chess where underpromotions to bishop and rook are forbidden" ++ No they are allowed, but underpromotions to pieces not previously captured not.
Now try reading what I wrote. I referred to generating drawing strategies for a MODIFIED game where underpromotions to bishop and rook are forbidden. These strategies would do for chess as well due to my valid reasoning. You should be pleased with this!
"The number of legal chess positions without ANY promotion is around 4 x 10^37."
++ Correction: without any promotion to a piece not previously captured.
No. My statement is true AND your statement is also true. They are the SAME set of positions.
For brevity the article title does not mention that, but it is obvious. Yes, to everyone.
"strongly believed that chess is a draw" ++ To me it is proven.
That is because, as has been widely observed, you don't know what a proof is (a key piece of knowledge early in a mathematical education).
"It's safe to say chess still can't be rigorously solved."
++ No, on the contrary it is safe to say chess can be rigourously weakly solved.
The 17 ICCF WC Finalist and their servers are doing it now.
110 games out of 110 that redundantly link the initial position in average 39 moves to certain draws.
You are remarkably clueless about what a weak solution is: it involves generating a complete proof tree. In a proof tree not a single legal move for the opposition is left unanalysed. Ponder on that a while.
Essential reading about the definition of proof tree (it applies to all of mathematics, not just weak solutions of games), In the weak solution of games the key step is that the game theoretic value (W/D/L) of a position is the maximum of the game theoretic values of the positions reachable by a legal move from that position.
And of course, you have no understanding of what rigor is. It's a shame this will never be fixed.
"Today's top chess engines will probably lose matches to those of the future."
++ At short time control. In TCEC they have to impose 50 unbalanced openings to avoid all draws and they do hit the 7-men endgame table base.
Current chess engines beat those of only a short time in the past in some games. It is absurd to suggest this must be over. The ICCF is a competition between Stockfish 16 and itself running on high end hardware. Even using significantly inferior hardware is enough to lose games. There is no reason to believe Stockfish 16 is invincible, just that there is nothing to beat it right now. Stockfish 17 will come, I assure you!
... each year there were fewer and fewer decisive games in the ICCF World Championship Finals and now there are none.
The thing is, this can have multiple causes.
The conclusion of "Therefore, we have reached perfect play, or close to it" (or some similar conclusion) isn't the only conclusion we can draw from the results.
Some other possible conclusions:
Possibility 1: Previous ICCF championships may have had greater disparities between competitors, in terms of hardware and software being used, and/or being available - thus leading to differences in playing strength among the competitors.
Possibility 2: ICCF competitors may have believed, in past years, that their human decisions were more valuable - thus they may have made more human choices in their games. But as the strength of engines took a large leap forward in recent years (due to AlphaZero's rise and the community acceptance of NNUE, for example), ICCF players may have learned to trust their hardware and software more, and to rely on their human input less and less - thus leading to a greater amount of draws, due to competitors all pitting the same hardware/software against each other with minimized human interference.
Possibility 3: ICCF players may have learned to embrace less volatile openings/defenses and have thus narrowed their repertoires to the safest options, to minimize their chances of losing ... resulting in a greater number of draws.
... We can consider more possibilities as well, if we just sit and think about it. And this is one of the necessary steps to avoid leaps in logic - we have to consider other possibilities and examine them accordingly ...
tygxc...i dont see any hard #'s coming from ur naysayers. do u ?...let alone a way to get there. they attack u but wheres THEIR answers ?...see ?...they wouldnt even know where to start in dev'ing the sw.
I am certain chess is a draw with optimal play, and I have justification.
I (and nearly everyone who plays chess) assumes chess is a draw with best play, and they can give reasonable arguments for that.
True. It should, of course, be a draw, due to how symmetrical the starting position is.
Even in 960, the two sides mirror each other.
This is why I've thought it would be far more interesting for a chess variant where each color has their starting position randomized (just as in Fischer Random) ... but each color's arrangement is independent of the other.
So we might have a starting position like this:
Then the players begin.
To make it fair, games are played in pairs, like in the TCEC - where the same setup is used for the next game, but the players switch sides ...
This would certainly reduce the number of draws, I believe ... and would make memorization and theory virtually irrelevant.
I'm sure others have thought of this (or a similar idea) as well ...
tygxc...i dont see any hard #'s coming from ur naysayers. do u ?...let alone a way to get there. they attack u but wheres THEIR answers ?...see ?...they wouldnt even know where to start in dev'ing the sw.
There's only one hard number in Tygxc's parade of premises...10^44, which is the same "hard number" everyone is using. Since the real answer is "there is no answer, nor one forthcoming anytime soon", your last point is meaningless.
Over a hundred new posts.
With @Optimissed losing again - as he always does.
As he saw he was losing again - he started talking about moderators and 'coded signals' ...
in other words went into his extra-paranoid mode that he always does as he's always defeated.
Similiar in the 'hoax' forum where he's overtly trying to get the opening poster there to block me.
And yes - BC doesn't have O's psychological handicap.
And as also always happens - O is defeated yet again by each and every person he throws stones at.
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tygxc is then at the center once more as usual and the actual forum topic is then addressed.
tygxc provides 'foils of illogic'.
Noting that tygxc is now using phrases that he is 'after the truth' ...
'the truth'.
Quite a buzz phrase on the net.
In math and in the logic of math - the numbers can't lie.
Numbers don't have that capability.
Of course numbers can be used to support falsehoods though.
Or illogic.
Illogic is not 'truth'.
More central to tygxc's positions is what he has decided 'matters'.
What he 'cares' about and sometimes he uses 'we'.
'We dont care about operations per second of computers'.
He's not going to let anybody tell him what he should prioritize.
Which is understandable and a very common thing.